Homological dimensions of modules of holomorphic functions on submanifolds of Stein manifolds

TL;DR

This paper investigates the homological dimensions of modules of holomorphic functions on submanifolds within Stein manifolds, establishing formulas relating these dimensions to geometric properties like codimension and types of the manifolds.

Contribution

It provides explicit formulas for the weak and projective homological dimensions of O(Y) as an O(X)-module, linking algebraic invariants to geometric features of Y and X.

Findings

Weak homological dimension equals codimension of Y in X.

For Liouville type manifolds, the projective dimension matches the weak dimension.

If X is Liouville type and Y is hyperconvex, the projective dimension equals the dimension of X.

Abstract

Let X be a Stein manifold, and let Y be a closed complex submanifold of X. Denote by O(X) the algebra of holomorphic functions on X. We show that the weak (i.e., flat) homological dimension of O(Y) as a Fr'echet O(X)-module equals the codimension of Y in X. In the case where X and Y are of Liouville type, the same formula is proved for the projective homological dimension of O(Y) over O(X). On the other hand, we show that if X is of Liouville type and Y is hyperconvex, then the projective homological dimension of O(Y) over O(X) equals the dimension of X.